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version 1.2, 1999/12/21 02:47:30 version 1.8, 2007/02/15 02:41:38
Line 1 
Line 1 
 @comment $OpenXM$  @comment $OpenXM: OpenXM/src/asir-doc/parts/algnum.texi,v 1.7 2003/04/20 08:01:24 noro Exp $
 \BJP  \BJP
 @node $BBe?tE*?t$K4X$9$k1i;;(B,,, Top  @node $BBe?tE*?t$K4X$9$k1i;;(B,,, Top
 @chapter $BBe?tE*?t$K4X$9$k1i;;(B  @chapter $BBe?tE*?t$K4X$9$k1i;;(B
Line 11 
Line 11 
 @menu  @menu
 \BJP  \BJP
 * $BBe?tE*?t$NI=8=(B::  * $BBe?tE*?t$NI=8=(B::
   * $BJ,;6B?9`<0$K$h$kBe?tE*?t$NI=8=(B::
 * $BBe?tE*?t$N1i;;(B::  * $BBe?tE*?t$N1i;;(B::
 * $BBe?tBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B::  * $BBe?tBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B::
 * $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B::  * $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B::
 \E  \E
 \BEG  \BEG
 * Representation of algebraic numbers::  * Representation of algebraic numbers::
   * Representation of algebraic numbers by distributed polynomials::
 * Operations over algebraic numbers::  * Operations over algebraic numbers::
 * Operations for uni-variate polynomials over an algebraic number field::  * Operations for uni-variate polynomials over an algebraic number field::
 * Summary of functions for algebraic numbers::  * Summary of functions for algebraic numbers::
Line 237  t#0
Line 239  t#0
 [100]  [100]
 @end example  @end example
   
   
   @example
   @end example
   
 \BJP  \BJP
 @node $BBe?tE*?t$N1i;;(B,,, $BBe?tE*?t$K4X$9$k1i;;(B  @node $BBe?tE*?t$N1i;;(B,,, $BBe?tE*?t$K4X$9$k1i;;(B
 @section $BBe?tE*?t$N1i;;(B  @section $BBe?tE*?t$N1i;;(B
Line 410  into the @b{root} by @code{rattoalgp()} function.
Line 416  into the @b{root} by @code{rattoalgp()} function.
   
 @example  @example
 [88] rattoalgp(S,[alg(0)]);  [88] rattoalgp(S,[alg(0)]);
 (((#0+2)/(#0+2))*t#1^2+((#0^2+2*#0)/(#0+2))*t#1+((2*#0^2+4*#0)/(#0+2)))*x  (((#0+2)/(#0+2))*t#1^2+((#0^2+2*#0)/(#0+2))*t#1
 +((1)/(#0+2))*t#1+((1)/(#0+2))  +((2*#0^2+4*#0)/(#0+2)))*x+((1)/(#0+2))*t#1+((1)/(#0+2))
 [89] rattoalgp(S,[alg(0),alg(1)]);  [89] rattoalgp(S,[alg(0),alg(1)]);
 (((#0^3+6*#0^2+12*#0+8)*#1^2+(#0^4+6*#0^3+12*#0^2+8*#0)*#1+2*#0^4+12*#0^3  (((#0^3+6*#0^2+12*#0+8)*#1^2+(#0^4+6*#0^3+12*#0^2+8*#0)*#1
 +24*#0^2+16*#0)/(#0^3+6*#0^2+12*#0+8))*x+(((#0+2)*#1+#0+2)/(#0^2+4*#0+4))  +2*#0^4+12*#0^3+24*#0^2+16*#0)/(#0^3+6*#0^2+12*#0+8))*x
   +(((#0+2)*#1+#0+2)/(#0^2+4*#0+4))
 [90] rattoalgp(S,[alg(1),alg(0)]);  [90] rattoalgp(S,[alg(1),alg(0)]);
 (((#0+2)*#1^2+(#0^2+2*#0)*#1+2*#0^2+4*#0)/(#0+2))*x+((#1+1)/(#0+2))  (((#0+2)*#1^2+(#0^2+2*#0)*#1+2*#0^2+4*#0)/(#0+2))*x
   +((#1+1)/(#0+2))
 [91] simpalg(@@89);  [91] simpalg(@@89);
 (#1^2+#0*#1+2*#0)*x+((-1/5*#0+2/5)*#1-1/5*#0+2/5)  (#1^2+#0*#1+2*#0)*x+((-1/5*#0+2/5)*#1-1/5*#0+2/5)
 [92] simpalg(@@90);  [92] simpalg(@@90);
Line 444  used for your own simplification.
Line 452  used for your own simplification.
 \E  \E
   
 \BJP  \BJP
   @node $BJ,;6B?9`<0$K$h$kBe?tE*?t$NI=8=(B,,, $BBe?tE*?t$K4X$9$k1i;;(B
   @section $BJ,;6B?9`<0$K$h$kBe?tE*?t$NI=8=(B
   \E
   \BEG
   @node Representation of algebraic numbers by distributed polynomials,,, Algebraic numbers
   @section Representation of algebraic numbers by distributed polynomials
   \E
   
   @noindent
   \BJP
   $BA0@a$G=R$Y$?$h$&$K(B, @code{root} $B$r4^$`Be?tE*?t$KBP$9$k4JC12=$O(B
   $B%f!<%6$NH=CG$G9T$&I,MW$,$"$k(B. $B$3$l$KBP$7(B, $B$3$3$G2r@b$9$k$b$&0l$D$NBe?tE*?t$N(B
   $BI=8=$K$D$$$F$O(B, $B2C8:>h=|(B, $B%Y%-$J$I$r9T$C$?$"$H<+F0E*$K4JC12=$,9T$o$l$k(B.
   $B$3$NI=8=$O(B, $BC`<!3HBg$N>l9g$KFC$K8zN($h$/7W;;$,9T$o$l$k$h$&@_7W$5$l$F$*$j(B,
   $B%0%l%V%J!<4pDl4X78$N4X?t$K$*$1$k78?tBN$H$7$FMQ$$$k$3$H$,$G$-$k(B. $B$3$NI=8=$O(B
   $BFbItE*$K$O(B, @code{DAlg} $B$H8F$P$l$k%*%V%8%'%/%H$H$7$FDj5A$5$l$F$$$k(B.
   @code{DAlg} $B$OJ,?t<0$N7A$GJ];}$5$l$k(B. $BJ,Jl$O@0?t(B, $BJ,;R$O@0?t78?t$NJ,;6B?9`<0$G$"$k(B.
   \E
   \BEG
   Simplification of algebraic numbers containing @code{root}
   is not done automatically and should be done by users.
   There is another representation of algebraic numbers, for which the
   results of fundamental operations are automatically simplified.
   This representations are designed so that operations are efficiently
   performed especially when the field is a successive extension and
   it can be used as a ground field for Groebner basis related functions.
   Internally an algebraic number of this type is defined as an object
   called @code{DAlg}. A @code{DAlg} is represented as a fraction. The
   denominator is an integer and the numerator is a distributed polynomial
   with integral coefficients.
   \E
   
   \BJP
   @code{DAlg} $B$O!$(B@code{set_field()} $B$K$h$j$"$i$+$8$a@_Dj$5$l$?Be?tBN$N(B
   $B85$H$7$F@8@.$5$l$k(B. $B@8@.J}K!$O(B, @code{newalg()} $B$G@8@.$5$l$?Be?tE*?t$r(B
   $B4^$`?t$+$i(B @code{algtodalg()} $B$GJQ49$9$k(B, $B$"$k$$$OJ,;6B?9`<0$+$iD>@\(B
   @code{dptodalg()} $B$GJQ49$9$k!$$N(B 2 $BDL$j$"$k(B.
   $B0lC6(B @code{DAlg} $B7A<0$KJQ49$5$l$l$P(B, $B1i;;8e$K<+F0E*$K4JC12=$5$l$k(B.
   \E
   \BEG
   @code{DAlg} is generated as an element of an algebraic number field
   set by @code{set_field()}. There are two methods to generate a @code{DAlg}.
   @code{algtodalg()} converts an algebraic number containing @code{root}
   to @code{DAlg}. @code{dptodalg()} directly converts a distributed polynomial to
   @code{DAlg}.
   \E
   @example
   [0] A=newalg(x^2+1);
   (#0)
   [1] B=newalg(x^3+A*x+A);
   (#1)
   [2] set_field([B,A]);
   0
   [3] C=algtodalg(A+B);
   ((1)*<<1,0>>+(1)*<<0,1>>)
   [4] C^5;
   ((-11)*<<2,1>>+(5)*<<2,0>>+(10)*<<1,1>>+(9)*<<1,0>>+(11)*<<0,1>>
   +(-1)*<<0,0>>)
   [5] 1/C;
   ((2)*<<2,1>>+(-1)*<<2,0>>+(1)*<<1,1>>+(2)*<<1,0>>+(-3)*<<0,1>>
   +(-1)*<<0,0>>)/5
   @end example
   \BJP
   $B$3$NNc$G$O(B, Q(a,b) (a^2+1=0, b^3+ab+b=0) $B$K$*$$$F(B, (a+b)^5 $B$*$h$S(B 1/(a+b) $B$r(B
   $B7W;;(B ($B4JC12=(B) $B$7$F$$$k(B. $BJ,;R$G$"$kJ,;6B?9`<0$NI=<($O(B, $BJ,;6B?9`<0$NI=<($r$=$N$^$^N.MQ$7$F$$$k(B.
   \E
   \BEG
   In this example Q(a,b) (a^2+1=0, b^3+ab+b=0) is set as the current ground field,
   and (a+b)^5 and 1/(a+b) are simplified in the field. The numerators of the results
   are printed as distributed polynomials.
   \E
   
   \BJP
 @node $BBe?tBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B,,, $BBe?tE*?t$K4X$9$k1i;;(B  @node $BBe?tBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B,,, $BBe?tE*?t$K4X$9$k1i;;(B
 @section $BBe?tBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B  @section $BBe?tBN>e$G$N(B 1 $BJQ?tB?9`<0$N1i;;(B
 \E  \E
Line 499  where the ground field is a multiple extension.
Line 580  where the ground field is a multiple extension.
 (#0)  (#0)
 [64] B=newalg(75*s^2+(10*A^7-175*A^4-470*A)*s+3*A^8-45*A^5-261*A^2);  [64] B=newalg(75*s^2+(10*A^7-175*A^4-470*A)*s+3*A^8-45*A^5-261*A^2);
 (#1)  (#1)
 [65] P1=75*x^2+(150*B+10*A^7-175*A^4-395*A)*x+(75*B^2+(10*A^7-175*A^4-395*A)*B  [65] P1=75*x^2+(150*B+10*A^7-175*A^4-395*A)*x
 +13*A^8-220*A^5-581*A^2)$  +(75*B^2+(10*A^7-175*A^4-395*A)*B+13*A^8-220*A^5-581*A^2)$
 [66] P2=x^2+A*x+A^2$  [66] P2=x^2+A*x+A^2$
 [67] cr_gcda(P1,P2,[B,A]);  [67] cr_gcda(P1,P2);
 27*x+((#0^6-19*#0^3-65)*#1-#0^7+19*#0^4+38*#0)  27*x+((#0^6-19*#0^3-65)*#1-#0^7+19*#0^4+38*#0)
 @end example  @end example
   
Line 531  The function to do this factorization is @code{asq()}.
Line 612  The function to do this factorization is @code{asq()}.
 [116] A=newalg(x^2+x+1);  [116] A=newalg(x^2+x+1);
 (#4)  (#4)
 [117] T=simpalg((x+A+1)*(x^2-2*A-3)^2*(x^3-x-A)^2);  [117] T=simpalg((x+A+1)*(x^2-2*A-3)^2*(x^3-x-A)^2);
 x^11+(#4+1)*x^10+(-4*#4-8)*x^9+(-10*#4-4)*x^8+(16*#4+20)*x^7+(24*#4-6)*x^6  x^11+(#4+1)*x^10+(-4*#4-8)*x^9+(-10*#4-4)*x^8+(16*#4+20)*x^7
 +(-29*#4-31)*x^5+(-15*#4+28)*x^4+(38*#4+29)*x^3+(#4-23)*x^2+(-21*#4-7)*x  +(24*#4-6)*x^6+(-29*#4-31)*x^5+(-15*#4+28)*x^4+(38*#4+29)*x^3
 +(3*#4+8)  +(#4-23)*x^2+(-21*#4-7)*x+(3*#4+8)
 [118] asq(T);  [118] asq(T);
 [[x^5+(-2*#4-4)*x^3+(-#4)*x^2+(2*#4+3)*x+(#4-2),2],[x+(#4+1),1]]  [[x^5+(-2*#4-4)*x^3+(-#4)*x^2+(2*#4+3)*x+(#4-2),2],[x+(#4+1),1]]
 @end example  @end example
Line 641  The function is @code{sp()}.
Line 722  The function is @code{sp()}.
   
 @example  @example
 [103] sp(x^5-2);  [103] sp(x^5-2);
 [[x+(-#1),2*x+(#0^3*#1^3+#0^4*#1^2+2*#1+2*#0),2*x+(-#0^4*#1^2),2*x  [[x+(-#1),2*x+(#0^3*#1^3+#0^4*#1^2+2*#1+2*#0),2*x+(-#0^4*#1^2),
 +(-#0^3*#1^3),x+(-#0)],[[(#1),t#1^4+t#0*t#1^3+t#0^2*t#1^2+t#0^3*t#1+t#0^4],  2*x+(-#0^3*#1^3),x+(-#0)],
 [(#0),t#0^5-2]]]  [[(#1),t#1^4+t#0*t#1^3+t#0^2*t#1^2+t#0^3*t#1+t#0^4],[(#0),t#0^5-2]]]
 @end example  @end example
   
 @noindent  @noindent
Line 706  may yield a polynomial which differs by a constant.
Line 787  may yield a polynomial which differs by a constant.
 * rattoalgp::  * rattoalgp::
 * cr_gcda::  * cr_gcda::
 * sp_norm::  * sp_norm::
 * asq af::  * asq af af_noalg::
 * sp::  * sp sp_noalg::
   * set_field::
   * algtodalg dalgtoalg dptodalg dalgtodp::
 @end menu  @end menu
   
 \JP @node newalg,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B  \JP @node newalg,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
Line 1074  substitutes a @b{root} for the associated indeterminat
Line 1157  substitutes a @b{root} for the associated indeterminat
 @findex cr_gcda  @findex cr_gcda
   
 @table @t  @table @t
 @item cr_gcda(@var{poly1},@var{poly2},@var{alist})  @item cr_gcda(@var{poly1},@var{poly2})
 \JP :: $BBe?tBN>e$N(B 1 $BJQ?tB?9`<0$N(B GCD  \JP :: $BBe?tBN>e$N(B 1 $BJQ?tB?9`<0$N(B GCD
 \EG :: GCD of two uni-variate polynomials over an algebraic number field.  \EG :: GCD of two uni-variate polynomials over an algebraic number field.
 @end table  @end table
Line 1083  substitutes a @b{root} for the associated indeterminat
Line 1166  substitutes a @b{root} for the associated indeterminat
 @item return  @item return
 \JP $BB?9`<0(B  \JP $BB?9`<0(B
 \EG polynomial  \EG polynomial
 @item poly1, poly2  @item poly1  poly2
 \JP $BB?9`<0(B  \JP $BB?9`<0(B
 \EG polynomial  \EG polynomial
 @item alist  
 \JP $B%j%9%H(B  
 \EG list  
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
Line 1098  substitutes a @b{root} for the associated indeterminat
Line 1178  substitutes a @b{root} for the associated indeterminat
 @item  @item
 \JP 2 $B$D$N(B 1 $BJQ?tB?9`<0$N(B GCD $B$r5a$a$k(B.  \JP 2 $B$D$N(B 1 $BJQ?tB?9`<0$N(B GCD $B$r5a$a$k(B.
 \EG Finds the GCD of two uni-variate polynomials.  \EG Finds the GCD of two uni-variate polynomials.
 @item  
 \BJP  
 @var{alist} $B$OF~NO$K8=$l$k(B @code{root} $B$*$h$S(B, $B$=$l$i$NDj5A$K4^$^$l$k(B  
 @code{root} $B$r:F5"E*$K<h$j=P$7$FJB$Y$?%j%9%H(B. @var{a} $B$,(B @var{b} $B$N(B  
 $BDj5A$K4^$^$l$F$$$k>l9g(B, @var{a} $B$O(B @var{b} $B$h$j8e(B ($B1&(B) $B$KJB$P$J$1$l$P(B  
 $B$J$i$J$$(B.  
 \E  
 \BEG  
 @var{alist} is a list of @b{root}'s.  
 All the @b{root}'s appearing in the input and those required to define  
 the @b{root}'s in the list  must appear in the list. In the list  
 ,if the defining polynomial of @var{a} contains @var{b}  
 then @var{a} must come first.  
 \E  
 @end itemize  @end itemize
   
 @example  @example
Line 1119  then @var{a} must come first.
Line 1185  then @var{a} must come first.
 [77] Y=x^6+6*x^5+24*x^4+8*x^3-48*x^2+384*x+1024$  [77] Y=x^6+6*x^5+24*x^4+8*x^3-48*x^2+384*x+1024$
 [78] A=newalg(X);  [78] A=newalg(X);
 (#0)  (#0)
 [79] cr_gcda(X,subst(Y,x,x+A),[A]);  [79] cr_gcda(X,subst(Y,x,x+A));
 x+(-#0)  x+(-#0)
 @end example  @end example
   
 @table @t  @table @t
 \JP @item $B;2>H(B  \JP @item $B;2>H(B
 \EG @item Reference  \EG @item Reference
 @fref{gr hgr gr_mod}, @fref{asq af}  @fref{gr hgr gr_mod}, @fref{asq af af_noalg}
 @end table  @end table
   
 \JP @node sp_norm,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B  \JP @node sp_norm,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
Line 1209  x^12+2*x^8+5*x^4+1
Line 1275  x^12+2*x^8+5*x^4+1
 @table @t  @table @t
 \JP @item $B;2>H(B  \JP @item $B;2>H(B
 \EG @item Reference  \EG @item Reference
 @fref{res}, @fref{asq af}  @fref{res}, @fref{asq af af_noalg}
 @end table  @end table
   
 \JP @node asq af,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B  \JP @node asq af af_noalg,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
 \EG @node asq af,,, Summary of functions for algebraic numbers  \EG @node asq af af_noalg,,, Summary of functions for algebraic numbers
 @subsection @code{asq}, @code{af}  @subsection @code{asq}, @code{af}, @code{af_noalg}
 @findex asq  @findex asq
 @findex af  @findex af
   @findex af_noalg
   
 @table @t  @table @t
 @item asq(@var{poly})  @item asq(@var{poly})
Line 1226  x^12+2*x^8+5*x^4+1
Line 1293  x^12+2*x^8+5*x^4+1
 algebraic number field.  algebraic number field.
 \E  \E
 @item af(@var{poly},@var{alglist})  @item af(@var{poly},@var{alglist})
   @itemx af_noalg(@var{poly},@var{defpolylist})
 \JP :: $BBe?tBN>e$N(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B  \JP :: $BBe?tBN>e$N(B 1 $BJQ?tB?9`<0$N0x?tJ,2r(B
 \BEG  \BEG
 :: Factorization of polynomial @var{poly} over an  :: Factorization of polynomial @var{poly} over an
Line 1243  algebraic number field.
Line 1311  algebraic number field.
 @item alglist  @item alglist
 \JP @code{root} $B$N%j%9%H(B  \JP @code{root} $B$N%j%9%H(B
 \EG @code{root} list  \EG @code{root} list
   @item defpolylist
   \JP @code{root} $B$rI=$9ITDj85$HDj5AB?9`<0$N%Z%"$N%j%9%H(B
   \EG @code{root} list of pairs of an indeterminate and a polynomial
 @end table  @end table
   
 @itemize @bullet  @itemize @bullet
Line 1280  In the second argument @code{alglist}, @b{root} define
Line 1351  In the second argument @code{alglist}, @b{root} define
 first.  first.
 \E  \E
 @item  @item
 \JP $B7k2L$O(B, $BDL>o$NL5J?J}J,2r(B, $B0x?tJ,2r$HF1MM(B [@b{$B0x;R(B}, @b{$B=EJ#EY(B}] $B$N%j%9%H$G$"$k(B.  \BJP
   @code{af(F,AL)} $B$K$*$$$F(B, @code{AL} $B$OBe?tE*?t$N%j%9%H$G$"$j(B, $BM-M}?tBN$N(B
   $BBe?t3HBg$rI=$9(B. @code{AL=[An,...,A1]} $B$H=q$/$H$-(B, $B3F(B @code{Ak} $B$O(B, $B$=$l$h$j(B
   $B1&$K$"$kBe?tE*?t$r78?t$H$7$?(B, $B%b%K%C%/$JDj5AB?9`<0$GDj5A$5$l$F$$$J$1$l$P(B
   $B$J$i$J$$(B.
   \E
 \BEG  \BEG
   In @code{af(F,AL)}, @code{AL} denotes a list of @code{roots} and it
   represents an algebraic number field. In @code{AL=[An,...,A1]} each
   @code{Ak} should be defined as a root of a defining polynomial
   whose coefficients are in @code{Q(A(k+1),...,An)}.
   \E
   
   @example
   [1] A1 = newalg(x^2+1);
   [2] A2 = newalg(x^2+A1);
   [3] A3 = newalg(x^2+A2*x+A1);
   [4] af(x^2+A2*x+A1,[A2,A1]);
   [[x^2+(#1)*x+(#0),1]]
   @end example
   
   \BJP
   @code{af_noalg} $B$G$O(B, @var{poly} $B$K4^$^$l$kBe?tE*?t(B @var{ai} $B$rITDj85(B @var{vi}
   $B$GCV$-49$($k(B. @code{defpolylist} $B$O(B, [[vn,dn(vn,...,v1)],...,[v1,d(v1)]]
   $B$J$k%j%9%H$G$"$k(B. $B$3$3$G(B @var{di}(vi,...,v1) $B$O(B @var{ai} $B$NDj5AB?9`<0$K$*$$$F(B
   $BBe?tE*?t$rA4$F(B @var{vj} $B$KCV$-49$($?$b$N$G$"$k(B.
   \E
   \BEG
   To call @code{sp_noalg}, one should replace each algebraic number
   @var{ai} in @var{poly} with an indeterminate @var{vi}. @code{defpolylist}
   is a list [[vn,dn(vn,...,v1)],...,[v1,d(v1)]]. In this expression
   @var{di}(vi,...,v1) is a defining polynomial of @var{ai} represented
   as a multivariate polynomial.
   \E
   
   @example
   [1] af_noalg(x^2+a2*x+a1,[[a2,a2^2+a1],[a1,a1^2+1]]);
   [[x^2+a2*x+a1,1]]
   @end example
   
   @item
   \BJP
   $B7k2L$O(B, $BDL>o$NL5J?J}J,2r(B, $B0x?tJ,2r$HF1MM(B [@b{$B0x;R(B}, @b{$B=EJ#EY(B}]
   $B$N%j%9%H$G$"$k(B. @code{af_noalg} $B$N>l9g(B, @b{$B0x;R(B} $B$K8=$l$kBe?tE*?t$O(B,
   @var{defpolylist} $B$K=>$C$FITDj85$KCV$-49$($i$l$k(B.
   \E
   \BEG
 The result is a list, as a result of usual factorization, whose elements  The result is a list, as a result of usual factorization, whose elements
 is of the form [@b{factor}, @b{multiplicity}].  is of the form [@b{factor}, @b{multiplicity}].
   In the result of @code{af_noalg}, algebraic numbers in @v{factor} are
   replaced by the indeterminates according to @var{defpolylist}.
 \E  \E
 @item  @item
 \JP $B=EJ#EY$r9~$a$?0x;R$NA4$F$N@Q$O(B, @var{poly} $B$HDj?tG\$N0c$$$,$"$jF@$k(B.  \JP $B=EJ#EY$r9~$a$?0x;R$NA4$F$N@Q$O(B, @var{poly} $B$HDj?tG\$N0c$$$,$"$jF@$k(B.
Line 1294  the input polynomial by a constant.
Line 1412  the input polynomial by a constant.
 @end itemize  @end itemize
   
 @example  @example
   [98] A = newalg(t^2-2);
   (#0)
 [99] asq(-x^4+6*x^3+(2*alg(0)-9)*x^2+(-6*alg(0))*x-2);  [99] asq(-x^4+6*x^3+(2*alg(0)-9)*x^2+(-6*alg(0))*x-2);
 [[-x^2+3*x+(#0),2]]  [[-x^2+3*x+(#0),2]]
 [100] af(-x^2+3*x+alg(0),[alg(0)]);  [100] af(-x^2+3*x+alg(0),[alg(0)]);
 [[x+(#0-1),1],[-x+(#0+2),1]]  [[x+(#0-1),1],[-x+(#0+2),1]]
   [101] af_noalg(-x^2+3*x+a,[[a,x^2-2]]);
   [[x+a-1,1],[-x+a+2,1]]
 @end example  @end example
   
 @table @t  @table @t
Line 1306  the input polynomial by a constant.
Line 1428  the input polynomial by a constant.
 @fref{cr_gcda}, @fref{fctr sqfr}  @fref{cr_gcda}, @fref{fctr sqfr}
 @end table  @end table
   
 \JP @node sp,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B  \JP @node sp sp_noalg,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
 \EG @node sp,,, Summary of functions for algebraic numbers  \EG @node sp sp_noalg,,, Summary of functions for algebraic numbers
 @subsection @code{sp}  @subsection @code{sp}, @code{sp_noalg}
 @findex sp  @findex sp
   
 @table @t  @table @t
 @item sp(@var{poly})  @item sp(@var{poly})
   @itemx sp_noalg(@var{poly})
 \JP :: $B:G>.J,2rBN$r5a$a$k(B.  \JP :: $B:G>.J,2rBN$r5a$a$k(B.
 \EG :: Finds the splitting field of polynomial @var{poly} and splits.  \EG :: Finds the splitting field of polynomial @var{poly} and splits.
 @end table  @end table
Line 1343  over the field.
Line 1466  over the field.
 @item  @item
 \BJP  \BJP
 $B7k2L$O(B, @var{poly} $B$N0x;R$N%j%9%H$H(B, $B:G>.J,2rBN$N(B, $BC`<!3HBg$K$h$kI=8=(B  $B7k2L$O(B, @var{poly} $B$N0x;R$N%j%9%H$H(B, $B:G>.J,2rBN$N(B, $BC`<!3HBg$K$h$kI=8=(B
 $B$+$i$J$k%j%9%H$G$"$k(B.  $B$+$i$J$k%j%9%H$G$"$k(B. @code{sp_noalg} $B$G$O(B, $BA4$F$NBe?tE*?t$,(B, $BBP1~$9$k(B
   $BITDj85(B ($BB($A(B @code{#i} $B$KBP$9$k(B @code{t#i}) $B$KCV$-49$($i$l$k(B. $B$3$l$K(B
   $B$h$j(B, @code{sp_noalg} $B$N=PNO$O(B, $B@0?t78?tB?JQ?tB?9`<0$N%j%9%H$H$J$k(B.
 \E  \E
 \BEG  \BEG
 The result consists of a two element list: The first element is  The result consists of a two element list: The first element is
 the list of all linear factors of @var{poly}; the second element is  the list of all linear factors of @var{poly}; the second element is
 a list which represents the successive extension of the field.  a list which represents the successive extension of the field.
   In the result of @code{sp_noalg} all the algebraic numbers are replaced
   by the special indeterminate associated with it, that is @code{t#i}
   for @code{#i}. By this operation the result of @code{sp_noalg}
   is a list containing only integral polynomials.
 \E  \E
 @item  @item
 \BJP  \BJP
Line 1359  a list which represents the successive extension of th
Line 1488  a list which represents the successive extension of th
 \E  \E
 \BEG  \BEG
 The splitting field is represented as a list of pairs of form  The splitting field is represented as a list of pairs of form
 @code{[root,algptorat(defpoly(root))]}.  @code{[root,} @code{algptorat(defpoly(root))]}.
 In more detail, the list is interpreted as a representation  In more detail, the list is interpreted as a representation
 of successive extension obtained by adjoining @b{root}'s  of successive extension obtained by adjoining @b{root}'s
 to the rational number field.  Adjoining is performed from the right  to the rational number field.  Adjoining is performed from the right
Line 1386  the builtin function @code{res()} is always used.
Line 1515  the builtin function @code{res()} is always used.
   
 @example  @example
 [101] L=sp(x^9-54);  [101] L=sp(x^9-54);
 [[x+(-#2),-54*x+(#1^6*#2^4),54*x+(#1^6*#2^4+54*#2),54*x+(-#1^8*#2^2),  [[x+(-#2),-54*x+(#1^6*#2^4),54*x+(#1^6*#2^4+54*#2),
 -54*x+(#1^5*#2^5),54*x+(#1^5*#2^5+#1^8*#2^2),-54*x+(-#1^7*#2^3-54*#1),  54*x+(-#1^8*#2^2),-54*x+(#1^5*#2^5),54*x+(#1^5*#2^5+#1^8*#2^2),
 54*x+(-#1^7*#2^3),x+(-#1)],[[(#2),t#2^6+t#1^3*t#2^3+t#1^6],[(#1),t#1^9-54]]]  -54*x+(-#1^7*#2^3-54*#1),54*x+(-#1^7*#2^3),x+(-#1)],
   [[(#2),t#2^6+t#1^3*t#2^3+t#1^6],[(#1),t#1^9-54]]]
 [102] for(I=0,M=1;I<9;I++)M*=L[0][I];  [102] for(I=0,M=1;I<9;I++)M*=L[0][I];
 [111] M=simpalg(M);  [111] M=simpalg(M);
 -1338925209984*x^9+72301961339136  -1338925209984*x^9+72301961339136
Line 1399  the builtin function @code{res()} is always used.
Line 1529  the builtin function @code{res()} is always used.
 @table @t  @table @t
 \JP @item $B;2>H(B  \JP @item $B;2>H(B
 \EG @item Reference  \EG @item Reference
 @fref{asq af}, @fref{defpoly}, @fref{algptorat}, @fref{sp_norm}.  @fref{asq af af_noalg}, @fref{defpoly}, @fref{algptorat}, @fref{sp_norm}.
 @end table  @end table
   
   \JP @node set_field,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
   \EG @node set_field,,, Summary of functions for algebraic numbers
   @subsection @code{set_field}
   @findex set_field
   
   @table @t
   @item set_field(@var{rootlist})
   \JP :: $BBe?tBN$r4pACBN$H$7$F@_Dj$9$k(B.
   \EG :: Set an algebraic number field as the currernt ground field.
   @end table
   
   @table @var
   @item return
   0
   @item rootlist
   \JP @code{root} $B$N%j%9%H(B
   \EG A list of @code{root}
   @end table
   
   @itemize @bullet
   @item
   \JP @code{root} $B$N%j%9%H(B @var{rootlist} $B$G@8@.$5$l$kBe?tBN$r4pACBN$H$7$F@_Dj$9$k(B.
   \BEG
   @code{set_field()} sets an algebraic number field generated by @code{root} in
   @var{rootlist} over Q.
   \E
   @item
   \BJP
   @code{root} $B$OFbItE*$K=g=x$E$1$i$l$F$$$k$N$G(B, @var{rootlist} $B$O=89g$H$7$F;XDj(B
   $B$9$l$P$h$$(B. ($B=g=x$O5$$K$7$J$/$F$h$$(B.)
   \E
   \BEG
   You don't care about the order of @code{root} in @var{rootlist}, because
   @code{root} are automatically ordered internally.
   \E
   @end itemize
   
   @example
   [0] A=newalg(x^2+1);
   (#0)
   [1] B=newalg(x^3+A);
   (#1)
   [2] C=newalg(x^4+B);
   (#1)
   [3] set_field([C,B,A]);
   0
   @end example
   
   @table @t
   \JP @item $B;2>H(B
   \EG @item Reference
   @fref{algtodalg dalgtoalg dptodalg dalgtodp}
   @end table
   
   \JP @node algtodalg dalgtoalg dptodalg dalgtodp,,, $BBe?tE*?t$K4X$9$kH!?t$N$^$H$a(B
   \EG @node algtodalg dalgtoalg dptodalg dalgtodp,,, Summary of functions for algebraic numbers
   @subsection @code{algtodalg}, @code{dalgtoalg}, @code{dptodalg}, @code{dalgtodp}
   @findex algtodalg
   @findex dalgtoalg
   @findex dpodalg
   @findex dalgtodp
   
   @table @t
   @item algtodalg(@var{alg})
   \JP :: $BBe?tE*?t(B @var{alg} $B$r(B @code{DAlg} $B$KJQ49$9$k(B.
   \EG :: Converts an algebraic number @var{alg} to a @code{DAlg}.
   @item dalgtoalg(@var{dalg})
   \JP :: @code{DAlg} @code{dalg} $B$rBe?tE*?t$KJQ49$9$k(B.
   \EG :: Converts a @code{DAlg} @code{dalg} to an algebraic number.
   @item dptodalg(@var{dp})
   \JP :: $BJ,;6B?9`<0(B @var{dp} $B$r(B @code{DAlg} $B$KJQ49$9$k(B.
   \EG :: Converts an algebraic number @var{alg} to a @code{DAlg}.
   @item dalgtodp(@var{dalg})
   \JP :: @code{DAlg} @code{dalg} $B$rJ,;6B?9`<0$KJQ49$9$k(B.
   \EG :: Converts a @code{DAlg} @code{dalg} to an algebraic number.
   @end table
   
   @table @var
   @item return
   \JP $BBe?tE*?t(B, @code{DAlg} $B$^$?$O(B [$BJ,;6B?9`<0(B,$BJ,Jl(B] $B$J$k%j%9%H(B
   \EG An algebraic number, a @code{DAlg} or a list [distributed polynomial,denominator]
   @item alg
   \JP  @code{root} $B$r4^$`Be?tE*?t(B
   \EG an algebraic number containing @code{root}
   @item dp
   \JP  $BM-M}?t78?tJ,;6B?9`<0(B
   \EG a distributed polynomial over Q
   @end table
   
   @itemize @bullet
   @item
   \JP @code{root} $B$r4^$`Be?tE*?t(B, @code{DAlg} $B$*$h$SJ,;6B?9`<04V$NJQ49$r9T$&(B.
   \BEG
   These functions are converters between @code{DAlg} and an algebraic number
   containing @code{root}, or a distributed polynomial.
   \E
   @item
   \BJP
   @code{DAlg} $B$,B0$9$Y$-Be?tBN$O(B, @code{set_field()} $B$K$h$j(B
   $B$"$i$+$8$a@_Dj$7$F$*$/I,MW$,$"$k(B.
   \E
   \BEG
   A ground field to which a @code{DAlg} belongs must be set by @code{set_field()}
   in advance.
   \E
   @item
   \BJP
   @code{dalgtodp()} $B$O(B, $BJ,;R$G$"$k@0?t78?tJ,;6B?9`<0$H(B, $BJ,Jl$G$"$k@0?t$rMWAG$K;}$D(B
   $B%j%9%H$rJV$9(B.
   \E
   \BEG
   @code{dalgtodp()} returns a list containing the numerator (a distributed polynomial)
   and the denominator (an integer).
   \E
   @item
   \BJP
   @code{algtodalg()}, @code{dptodalg()} $B$O4JC12=$5$l$?7k2L$rJV$9(B.
   \E
   \BEG
   @code{algtodalg()}, @code{dptodalg()} return the simplified result.
   \E
   @end itemize
   
   @example
   [0] A=newalg(x^2+1);
   (#0)
   [1] B=newalg(x^3+A*x+A);
   (#1)
   [2] set_field([B,A]);
   0
   [3] C=algtodalg((A+B)^10);
   ((408)*<<2,1>>+(103)*<<2,0>>+(-36)*<<1,1>>+(-446)*<<1,0>>
   +(-332)*<<0,1>>+(-218)*<<0,0>>)
   [4] dalgtoalg(C);
   ((408*#0+103)*#1^2+(-36*#0-446)*#1-332*#0-218)
   [5] D=dptodalg(<<10,10>>/10+2*<<5,5>>+1/3*<<0,0>>);
   ((-9)*<<2,1>>+(57)*<<2,0>>+(-63)*<<1,1>>+(-12)*<<1,0>>
   +(-60)*<<0,1>>+(1)*<<0,0>>)/30
   [6] dalgtodp(D);
   [(-9)*<<2,1>>+(57)*<<2,0>>+(-63)*<<1,1>>+(-12)*<<1,0>>
   +(-60)*<<0,1>>+(1)*<<0,0>>,30]
   @end example
   
   @table @t
   \JP @item $B;2>H(B
   \EG @item Reference
   @fref{set_field}
   @end table
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